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Metamath Proof Explorer

Theorem nfralw

Description: Bound-variable hypothesis builder for restricted quantification. Version of nfral with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 1-Sep-1999) Avoid ax-13 . (Revised by GG, 10-Jan-2024) (Proof shortened by Wolf Lammen, 13-Dec-2024)

Ref Expression
Hypotheses nfralw.1 
|- F/_ x A
nfralw.2 
|- F/ x ph
Assertion nfralw 
|- F/ x A. y e. A ph

Proof

Step Hyp Ref Expression
1 nfralw.1 
 |-  F/_ x A
2 nfralw.2 
 |-  F/ x ph
3 1 nfcri 
 |-  F/ x y e. A
4 3 nf5ri 
 |-  ( y e. A -> A. x y e. A )
5 2 nf5ri 
 |-  ( ph -> A. x ph )
6 4 5 hbral 
 |-  ( A. y e. A ph -> A. x A. y e. A ph )
7 6 nf5i 
 |-  F/ x A. y e. A ph